Based on Act 3, Scene 4 of Hamlet, did Gertrude know about the murder?

The text suggests that Gertrude has no idea what Hamlet is referring to when he talks about the murder of a king. Hamlet is confrontational and rude from the outset of their conversation. His mother is shocked by his harsh tone and fears for her life. When it seems as if Hamlet is going to assault her, she cries out. Her desperate cry leads to Polonius (who is hiding behind the arras) also crying out. Hamlet stabs him through the arras, killing him.

When Gertrude cries out that Hamlet has committed "a most rash and bloody deed," he compares what he has just done to the act of killing a king and marrying his brother. He alludes to what his father's ghost had told him about its own demise when it stated that Claudius poisoned it and then married Gertrude and assumed the throne. Hamlet says:

A bloody deed! almost as bad, good mother,
As kill a king, and marry with his brother.

Gertrude's shocked response indicates that she does not know what Hamlet is talking about:

As kill a king!

Her question emphasizes her lack of knowledge:

What have I done, that thou darest wag thy tongue
In noise so rude against me?

Hamlet's overly dramatic and metaphoric reply confuses her even more and she seeks further information:

Ay me, what act,
That roars so loud, and thunders in the index?

Hamlet verbally lashes out at his mother and severely reprimands her for having married one so unworthy of filling his father's shoes. He rants about the fact that she could not have been driven by love and says that she should be ashamed of what she has done for she was driven by lust.

Gertrude is, of course, aghast at Hamlet's assertions and seems to realize how much grief she has brought him. She asks him to stop speaking and acknowledges some guilt.

O Hamlet, speak no more:
Thou turn'st mine eyes into my very soul;
And there I see such black and grained spots
As will not leave their tinct.

Hamlet is relentless, however, and uses foul and lewd descriptions of his mother's deeds. Gertrude is overwhelmed by her son's damning tirade and begs him to stop. It is only when the ghost intervenes and tells him to comfort his mother that Hamlet softens somewhat. When he beseeches her to confess her sins, Gertrude tells him that he has broken her heart with his claims.

Hamlet later employs a different approach. He asks her to first refuse to sleep with Claudius for a few nights. She should then, when he cannot tolerate her absence any longer, woo him and convince him that he (Hamlet) is not mad, but crafty. Gertrude responds by telling him that she would be incapable of uttering a word of what her son has told her. 

It is ironic later when Gertrude speaks to Claudius and he asks about her son's condition that she tells him:

Mad as the sea and wind...

The fact that Gertrude was so distraught and overcome by Hamlet's assertions makes one believe that she has no knowledge of what Claudius has done. She gives no inkling that she was involved in any way--either directly or indirectly.   

If you are referring to the murder of Hamlet Sr., it is certainly not stated explicitly that she knew anything about it or even suspected it.  Hamlet makes some reference to it, calling Claudius a murderer but does not go into it further.  Of course, Gertrude doesn't call him on this either so perhaps she is in some ways indicating that she understands Hamlet's feelings about it?

When Hamlet refers to Claudius as "A murderer and a villain, A slave that is not twentieth part the tithe of your precent Lord," Gertrude says only "No more." but it would be difficult to construe this as an indication that she was aware of Hamlet Sr.'s murder.

Did Queen Gertrude know about Old Hamlet's murder and the identity of the murderer?

I agree that it is difficult to tell from the dialogue of the accusation. "To kill a King?!" The actor might have great latitude in how to act this line. The queen might be fretting over the discovery of her knowledge, or she might be genuinely shocked at the notion. Gertrude is an intriguing character. Lady MacBeth's schemes are clear and blood curdling. Gertrude may be played as cool and calculating, or she may be an empty headed pawn. I believe Shakespeare was allowing for this to be a highly interpreted character and the participation of both the actor and the audience fills that void. This lacuna invites this very question, among others.

It's impossible precisely to say. There is a case to be made in the text for either possibility.

To argue that she did know about Old Hamlet's murder, people turn to the evidence that her marriage to Claudius, she says, is "o'erhasty" - as if they should have waited to allow suspicion to subside? You can also play her reactions in the play scene as deeply uncomfortable ("the lady doth protest too much").

To argue that she didn't know, productions (such as the recent staging by Sir Trevor Nunn at London's Old Vic Theatre) simply have to play Gertrude as not very bright, and not very knowing. So Gertrude is horrified at the play, but doesn't put two and two together, and - potentially - only realises during the closet scene (Act 3, Scene 4), which then (in some productions) leads to her deliberately drinking the poison in the final scene.

Even the ghost is ambiguous - is she to go to heaven, or go the thorny way:

Taint not thy mind, nor let thy soul contrive
Against thy mother aught. Leave her to heaven,
And to those thorns that in her bosom lodge
To prick and sting her.

You can do it both ways. If you wanted a definite answer, it depends on how you read the closet scene, and Gertrude's behaviour after it. When Hamlet calls Claudius "a murderer and a villain" - does she understand what he means, specifically? Does she stop the "bloat king" Claudius from tempting her to bed after that scene? Shakespeare doesn't specify. It's up to your own reading of the text to make an argument for what Gertrude really knew.